Sequence is probably finite and the list of 54 integers shown is probably complete (all integers <= 10000 eventually iterate to a cycle based on these 54, although for example 111 takes 2635 iterations to do so via a maximum of 99889)

An equivalent conjecture is that all numbers converge to the trajectory of 1 shown in A119031. - N. J. A. Sloane, Aug 02 2009.